Sunnyvale Public Library

We investigate the structure of discontinuities in clearance (or minimum time) functions for nonlinear control systems with general, closed obstacles (or targets). We establish general results regarding interactions between admissible trajectories and clearance discontinuities: e.g. instantaneous increases in clearance when passing through a discontinuity, and propagation of discontinuity along optimal trajectories. Then, investigating sufficient conditions for discontinuities, we explore a common directionality condition for velocities at a point, characterized by strict positivity of the minimal Hamiltonian. Elementary consequences of this common directionality assumption are explored before demonstrating how, in concert with corresponding obstacle configurations, it gives rise to clearance discontinuities both on the surface of the obstacle and propagating out into free space. Minimal assumptions are made on the topological structure of obstacle sets.

While building GUIs programmatically gives you more control over the design process, some development tasks may require a quicker approach. Fortunately, Qt provides a great interface for arranging widgets and designing main windows, widgets, or dialogs. The graphical development tool, Qt Designer, is filled with widgets and other tools for building GUIs. With the application's drag-and-drop interface, you are able to create and customize your own Qt or PyQt applications.

This second edition explores GUI application development from the ground up with simple projects that teach the fundamentals of using PyQt6.

While most development tasks can be solved with buttons, text editing widgets, and other components provided by PyQt, you may at some point find yourself in a situation where no single widget provides you with the tools or functionality that you need. You might even find yourself needing to use a widget you made in other GUIs and therefore need a way to easily import your custom-made widget into other applications. Thankfully, PyQt allows developers to build and import their own widgets for solving new and unforeseen tasks.

Graphical user interfaces nowadays can be found on a multitude of devices, including desktop computers, mobile devices, and small touchscreens controlled by microcontrollers. While UIs are designed to fit the functional and technical needs of each device, the demands of these platforms have inspired The Qt Company to continue to build a set of scalable, polished, dynamic, and visually stunning tools for building user interfaces.